Download On integers with many small prime factors

C OMPOSITIO M ATHEMATICA
R. T IJDEMAN
On integers with many small prime factors
Compositio Mathematica, tome 26, no 3 (1973), p. 319-330
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COMPOSITIO MATHEMATICA, Vol. 26, Fasc. 3, 1973, pag. 319-330
Noordhoff International Publishing
Printed in the Netherlands
ON INTEGERS WITH MANY SMALL PRIME FACTORS
by
R.
Tijdeman
1 n2 ... be the sequence of
Let p be a prime, p ~ 3, and let n 1
all positive integers composed of primes p. It follows from a theorem
of Stormer [22], that
=
This result was improved upon by Thue [23 ]. He derived from his famous
result on the approximation of algebraic numbers by rationals that
See also Pôlya [15]. Siegel’s well known improvements [18, 19] of Thue’s
result made it possible to improve upon (1). Such an improvement can
be obtained immediately from a result of Mahler [13, 1, Satz 2]. But, as
far as 1 know, it was not before 1965 that such a result was stated explicitly in literature. In a survey paper [7, p. 218]Erdôs gave the following
improvement of (1). Let 0 8- 1. Then
In this paper we give several improvements and generalizations of inequality (2). In our Theorems 1 and 2 the problem of filling the gap between (2) and the opposite result limi~~ni+1/ni
1 (cf. [15]) is solved
almost completely. We shall prove that there exists an effectively computable constant C C(p) such that
=
=
while we shall show that this inequality is false for C
n(p)- 1. (Here
03C0(x) denotes the number of primes ~ x.)
It follows from (1) that for every integer k ~ 1 there exists a number
Akp such that ni + 1- ni &#x3E; k for ni &#x3E; Akp. Because of its ineffective nature it is impossible to obtain estimates for Akp from the methods of Thue
and Siegel. Using a result of Gel’fond, Cassels [5] gave the first "effective
proof" of (1). In 1964/65 D.H. Lehmer [11, 12] gave explicit upper bounds
319
320
for A1p, A2p and A4. for all p. In the Theorems 3 and 4 upper bounds are
given for Akp for all k and all p and also for N~ in (2). A consequence
b have the same
of this result is that if two integers a and b with 0 a
the
10-6
then
b
a
~
factor,
inequality
log log a holds.
greatest prime
Professor Straus suggested to me a generalization of (2) in another
direction, namely for integers ni which have many prime factors ~ p,
but are not necessarily composed of primes ~ p. Such a result is given by
Theorem 6.
Finally we solve a conjecture of Wintner [7, p. 218]in the afhrmative.
We prove even much more in Theorem 7:
1. There exists an infinite sequence of primes p,
Let 0 ~
P2 ...
such that, if ni
...
is
set
the
all
n2
of
integers composed of these
primes, then
Recent results proved by the beautiful Gel’fond-Baker method form the
basic tools for nearly all proofs in this paper. The results used for the
proofs of Theorems 1-5 have all been published. Since it is our main interest to indicate the possibility of the applications, we did not bother too
much to get as good results as possible, but it should be noted that it is
possible to improve upon our results by using not yet published results
of Stark and others, or by improving upon the auxiliary results. This
is possible because of additional information available here. For the
proofs of Theorems 6 and 7 a not yet published result of Baker [3] has
been used. This result is cited in section 7. I wish to express my thanks to
Dr. A Baker for communicating the exact formulation of his result to me.
1. As usual 03C0(x) denotes the number of primes ~ x. The number of
distinct prime factors and the greatest prime factor of a positive integer
a are denoted by 03C9(a) and P(a), respectively. For the sake of brevity we
write e.c. for effectively computable. In this paper all constants c, cl , C2,..
and C, C1, C2, ... are e.c.
2. We
use a
result of Fel’dman to prove
THEOREM 1. Let a and b be positive integers, 3
p
P(ab). Then
a
b. Put r
=
cv(ab),
=
where
Ci =
cr41(log p)14r2 and Ci is an e.c. absolute constant.
PROOF. Without loss of generality we may assume b ~ 2a and r ~ 2.
Let a 03A0rj=1p03B1jj and b
03A0rj=1p03B2jj be prime factorizations of a and b.
=
=
321
Hence
According to [9, Theorem 1 ] the following estimate holds :
where
and
c
is
some e.c.
absolute constant. Hence
This proves the theorem.
COROLLARY. Let ni
of primes not greater
n2 ... be the sequence of integers composed
than p. Then there exists an e.c. constant C
C(p)
=
such that
PROOF. We have r ~
4pJ(3Iogp). (See e. g. [17, formula (3.6)]or [10])
3. The
following theorem shows that the constants C1 and C in the
preceding section cannot be replaced by constants smaller than r -1 and
n(p )-1, respectively. The gap meant by Erdôs at [7,p. 128 ] is therefore
filled up almost completely.
THEOREM 2. Let P = {p1, ···, pr} be a given set of primes, r &#x3E; 1, and
put p = maxjpj. Then there exist infinitely many pairs of integers a, b
such that
PROOF. Let T be
positive integer and consider all numbers of the form
t1 log p1 + t2 log p2 + ··· +tr log pr with 0 ~ tj ~ T for j 1, ···, r.
We have (T+1)r non-negative numbers of absolute value ~ rT log p.
It follows that there are two among them, al log p1 + ··· + ocr log pr and
fli log p1 + ··· + 03B2r log pr say, such that
a
=
322
Hence,
r.
0 for j
Without loss of generality we may assume that cejflj
1,
Putting a 03B203B1 1
prr and b poi ... p03B2rr, we obtain a2 ~ ab ~ prT .
Hence, T ~ 2 log al(r log p). Substituting this estimate in (5) we see that
=
=
=
We know from
...
...,
=
(5) that log (bla)
-
0, if T -
The inequality
oo.
Hence,
(4) follows from (6) and (7). By letting T
infinitely many pairs a, b subject to (4).
-
oo
we
obtain
4. One cannot deduce an explicit bound for the N~ in (2) from the
proof of Siegel. The following proof based on the Gel’fond-Baker theory
shows that N~ can be chosen such that
THEOREM 3. Let 0
8
1. Let
a
and b be positive integers such that
PROOF. We have a &#x3E; 1 and r ~ 2. Let a
03A0rj=1p03B1jj and b
be prime factorizations of a and b respectively. Hence,
=
=
03A0rj=1p03B2jj
We have
and therefore
where 03B4 = ~/3 and H maxj
well known result [1] that
=
(03B2j-03B1j) ~ 3 log a. It follows from Baker’s
323
Hence,
This proves the theorem. We recall that r ~
4p/(3 log p).
5. The next theorem gives an upper bound for a similar to the one in
Theorem 3. The present estimate is better if b - a is rather small and r
is large. A ysimilar method of proof was given by Ramachandra [16,
Theorem 3]. Here we use Baker’s result on the integer solutions of the
diophantine equation y2 = X3 + k.
THEOREM 4. Let a and b be positive
p
P(ab). Then
integers,
a
b. Put
r
=
ce(ab)
and
=
p03B211 ··· 0" be prime factorizations
respectively. Let a ve3 and b wf2, where e,f, v, w are
positive integers such that v is cubefree and w is squarefree. We have
PROOF. Let
of
a
=
pil
...
prr and b
and b
a
=
=
=
Thus (x, y) = (vwe, vw2f) is an integer solution of the diophantine equation y2 x3+k, where k (b - a)v2w3. It follows from Baker’s estimate
=
=
for such solutions
We
[2], that
have |v| ~ p2r,|w| ~ pr. Hence,
which
implies
So
obtain
we
our
assertion.
COROLLARY. Let k and p be positive integers. Let Akp be the smallest
integer a such that if b - a k and P(ab) p then a ~ Akp. Then
=
=
324
proved for example log A1p ~ c2p2eP!2 for some absolute constant c2 and all p. He also included tables of numerical results for p ~ 41.
Akp .
He
It appears that the upper bounds for Akp are rather crude. It would be
of great interest to replace the factor ep by some constant power of p.
This would be possible, if Hall’s conjecture [4] on the solutions of y2 =
x3+k were proved. (M. Hall Jr. has conjectured that if x, y are integers
with x3 ~ y2, then |x3-y2| &#x3E; |x| for x &#x3E; xo . )
6. The following theorem has direct consequences for the number theoretic function f3(n) introduced by Erdôs and Selfridge [8 ]. In their notation it implies f3(n) &#x3E; 10 - 6 loglog n, which is the first non-trivial lower
bound for f3(n). They conjectured f3 (n) &#x3E; (log n)’3 for some absolute
constant C3 and all n. (This would also be a consequence of Hall’s con-
jecture.)
THEOREM 5. Let a and b be positive
P(a) P(b). Then
integers such that a
b and
=
PROOF. We know from the above
Hence, by P(a)
This
Corollary that
~ 2,
implies (i). Since P(a) ~ b-a,
we
have
7. In this section we generalize (2) in a direction which was suggested
by Professor E. G. Straus. As Professor Baker pointed out in a discussion
with Professor Straus the following recent result of his makes it possible
to give an inequality like (2) for integers containing many small prime
factors.
If 03B11, ···, an are non-zero algebraic numbers with degrees at most d,
if the heighths of 03B11, ···, 03B1n-1 and an are at most A’ and A(~ 2),
respectively, then there is an effectively computable number C, depending
only on n, d and A’ such that, for any ô with 0 ô 2, the inequalities
and
have no solution in rational integers b1, ···,
solute values at most B and B’, respectively.
hn -1
and
hn( =F 0)
with ab-
325
Using this result one can deduce results related to the p-adic analogues of
the theorems of Thue-Siegel-Roth and Baker as developed by Mahler,
Coates, Sprindzuk and others. (See for example [13, 14, 6, 20, 21].) We
note that one of the related conjectures at the end of Mahler’s book [14,
Appendix III] was proved by Straus, but that this result has not been
published. We prove here
8
THEOREM 6. Let 0
a
1. Let
a
and b be positive integers such that
b ~ a+a1-~. Furthermore,letp,r,a1,
such that
exists
a
al a2, b
=
an e.c.
a2, b1, b2 be p ositive integers
bi b2, P(al b1) ~ p, 03C9(a1 b1) ~ r. Then there
=
constant il
=
il(p,
r,
8)
&#x3E;
0 such that
PROOF. Put A
max (a2, b2). It suffices to prove A &#x3E; a’’ -1. First we
assume A ~ 2. Let al = pii ...prr, and bl p03B211 ··· p03B2rr be prime fac=
=
torizations
of al
and
b1 respectively. Hence,
We have |03B2j-03B1j| ~ log b/log 2 for j = 1, ..., r. We applythe above mentio(11). We take
ned result of Baker to the linear form at the right hand side of
B
=
log b/log 2 and.B’
=
1.
Hence, there exists an e.c. constant C2
such that
Since b ~ 2a, it follows that
On the other
hand,
The combination of these
or,
equivalently,
inequalities gives
=
C2 (p,r)
326
Let 1
0 be chosen
&#x3E;
small that
so
Then a2 ~ 2A1/~. Hence A ~
a’’ .
Now we consider the remaining case A = 1. We see from Theorem 3
that there exists an e.c. constant C3 - C3(p, r, 9) &#x3E; 0 such that a
C3.
We choose q &#x3E; 0 so small that (12) is fulfilled and, moreover,
Hence, A = 1 &#x3E; C~3 -1 &#x3E; a~-1. This completes the proof of the
theorem.
8. It is easy to
see
that Theorem 6 is
no
longer valid, if we replace (10)
9. Finally we give another application of the result of Baker mentioned
in 7. Wintner communicated the following problem to Erdôs orally
[7, p. 218].
Does there exist an infinite sequence of primes p 1
p2 ···
such that if n1 n2 ··· is the set of all integers composed of the p’s
then limi~~(ni+1-ni)
oo?
Here we solve the conjecture in the affirmative and prove even much more:
=
THEOREM 7. Let 0
~
1. There exists an infinite sequence of primes
such
that
p2
if ni n2 ... is the set of all integers comthese
then
posed of
primes
...
Pl
PROOF. We construct
a
sequence pl
p2
with the required prop(13) holds for powers of
...
p, such that
...
erty by induction. Let p1 = 3. It is obvious that
3. Now
assume
for all
that
we
have
a
sequence p1
b and composed of these primes. Let p be
that there exist integers a and b composed
of the primes p1,...,pr, p such that 0 b - a a1-~. Let a = p03B111 ···
p03B1rrp03B1 and b Pol ...p03B2rrp03B2 be prime factorizations of a and b, respectively. Then 11 1= P because of our induction hypothesis applied to alp’
and b/p03B2. We have
a
integers a, b with a
prime, p &#x3E; pr. Suppose
=
327
Furthermore, |03B2j-03B1j| ~ log b ~ log 2a for j 1,···, r and |03B2-03B1| ~
log b/log p ~ log (2a)/log p. We apply Baker’s result (see 7) with d 1,
n
r+ 1,A’ - pr, A p, b = 814, B log (2a) and B’ = log (2a)l
log p. Hence, there exists an e.c. constant C4 C4(p, r) such that
=
=
=
=
=
=
It follows from
This
(15) and (16) that
implies
C5 - CS(p, r, 8) and C6 = C6(8) such
that x ~ CS log (C6 x), where x log a/log p. Hence, there exists an e.c.
constant C7 = C7(p, r, 8) such that
Thus there exist
e.c.
constants
=
obtain a ~ pC7 and b ~ 2a ~ pC7, whence 03B1, 03B2 ~ C7 and
03B1j, 03B2j ~ C7 log p for j = 1, ···, r.
Let T be a large integer. We know from [17, formula (3.8)] that the
number of primes in the interval [T/2, T] is larger than 3T/(10 log T)
for T ~ 41. For each prime p in the interval [T/2, Twe apply the above
argument. Hence,
So
we
implies that a, j8 ~ C7 and 03B1j, 03B2j ~ C7 log T for j 1, - - -, r. The number of possible choices for 03B11, ···, a,. , a, 03B21, ···, f3r, f3 is at most
(C7+ 1)2r+2(log T) 2r. Assume that all these integers are fixed. Then
1
b/a 1 +a-8 implies that
=
where
Then
C8
=
p03B11-03B211 ···p03B1r-03B2rr. We
recall that 03B1 ~
Hence, p is contained in a fixed interval of length
03B2. First suppose p
&#x3E; oc.
328
Secondly suppose fi
11.
Hence, p is contained in
Then from
a
(18)
fixed interval of length
the number of primes p for which (17)
with fixed 03B11, ···, ar, a, Pl, - - -, fi,, fl is possible does not exceed Ta-~.
Since 2a ~ b ~ p ~ T/2, we see that Ta-~ ~ 4T1-~. We conclude
that the total number of primes in the interval [T/2, T] for which integers
a and b subject to (17) can be found is at most
It turns out that in both
cases
We have to exclude these primes. However, there are more than
3T/(10 log T ) primes in this interval. For sufficiently large Tthe number of
primes in [T /2, Tis greater than (19) and we can take pr + 1 out of the
remaining set. Doing so every pair of integers a and b with a b and
composed of p1, ···, pr+1 satisfies b - a &#x3E; al - ~. Now the proof has been
completed by induction.
10. REMARKS. (i) It follows from the above proof that for every C
with 0 9
1 it is possible effectively to give a sequence Pl, P 2’ ...
such that there exists a sequence p1
p2 ... with the required
1. It follows
property and with Pj/2 ~ pj ~ Pj for all j. (ii) 0 ~
from Theorem 2 that there does not exist a constant C9 = C9(~) such
that Theorem 7 is valid if (13) is replaced by the inequality
It is interesting to compare this with Theorem 1.
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=
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